combined qp - fp3 edexcelpmt.physicsandmathstutor.com/download/maths/a-level/fp3/topic-qs...10 ....
Post on 27-Mar-2018
235 Views
Preview:
TRANSCRIPT
PhysicsAndMathsTutor.com
Leave blank
10
*M35145A01028*
4. Given that y = arsinh (√x), x > 0,
(a) find d
d
y
x, giving your answer as a simplified fraction.
(3)
(b) Hence, or otherwise, find
11
4
[ ( )]x xx
+∫ 14
d ,
giving your answer in the form ln a b+⎛⎝⎜
⎞⎠⎟
52
, where a and b are integers. (6)
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
√
√
PhysicsAndMathsTutor.com June 2009
Leaveblank
12
*M35145A01228*
5.I x
xn
n
=−∫ ( )25 2
0
5
dx , n 0
(a) Find an expression for xx
x( )25 2−∫ d , 0 x 5.
(2)
(b) Using your answer to part (a), or otherwise, show that
(5)
(c) Find 4I in the form kπ, where k is a fraction. (4)
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
225( 1)
nn In n 2nI
Leaveblank
13
*M35145A01328* Turn over
Question 5 continued
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
Leaveblank
24
*M35145A02428*
8. A curve, which is part of an ellipse, has parametric equations
x y= =3 5cos , sin θ θ , 02
� �θ π .
The curve is rotated through 2 radians about the x-axis.
(a) Show that the area of the surface generated is given by the integral
k , where c = cos ,
and where k and are constants to be found.(6)
(b) Using the substitution c u= 34
sinh , or otherwise, evaluate the integral, showing all of
your working and giving the final answer to 3 significant figures.(5)
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
( )16 92
0c c+ d∫
Leaveblank
28
*M35145A02828*
Question 8 continued
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
TOTAL FOR PAPER: 75 MARKS
END
Q8
(Total 11 marks)
Leaveblank
4
*N35389RA0428*
2. Use calculus to find the exact value of1
22
1 d4 13
x .x x− + +∫ (5)
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
Leaveblank
8
*N35389RA0828*
4. I0
a
n ∫ ( – ) cos d , > 0, 0
(a) Show that, for 2n , = – 1 – ( – 1) – 2
(5)
(b) Hence evaluate
π
π−
−
2
22
x0
cos d .(3)
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
Leaveblank
9
*N35389RA0928* Turn over
Question 4 continued
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
Leave blank
2
*P35414A0228*
1. The curve C has equation 32 ,y x= 0 2x . The curve C is rotated through 2 radians about the x-axis. Using calculus, find the area of the surface generated, giving your answer to 3 significant
figures.(5)
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
Leave blank
6
*P35414A0628*
3. Show that
(a) 1
10 3425
8
x xx k
− +=∫ d π , giving the value of the fraction k,
(5)
(b) 8
25
1d ln( ),
( 10 34)x A n
x x= +
− +∫ giving the values of the integers A and n.(4)
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
Leave blank
8
*P35414A0828*
4.I x x x nn
n= ∫ 2
1
0(ln ) ,d e
�
(a) Prove that, for 1n ,
I n In n= − − e3
13 3 (4)
(b) Find the exact value of 3I . (4)
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
Leave blank
9
*P35414A0928* Turn over
Question 4 continued
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
Leave blank
4
*P40111A0432*
2.
Figure 1ln a x
y
O
The curve C, shown in Figure 1, has equation
y x= 13
3cosh , 0 � x ��ln a
where a is a constant and a > 1
Using calculus, show that the length of curve C is
k aa
( (33
1−
and state the value of the constant k. (6)
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
Leave blank
10
*P40111A01032*
4. I x x xnn= ∫ sin 2
0
4 dπ
, n� 0
(a) Prove that, for n� 2,
I n n n In
n
n= ⎛⎝⎜
⎞⎠⎟
− −−
−14 4
14
11
2 π ( )
(5)
(b) Find the exact value of I2(4)
(c) Show that I431
6424 48= − +( )π π
(2)
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
Leave blank
11
*P40111A01132* Turn over
Question 4 continued
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
Leave blank
14
*P40111A01432*
5. (a) Differentiate x x arsinh2 with respect to x. (3)
(b) Hence, or otherwise, find the exact value of
0
22∫ arsinh dx x
giving your answer in the form A B Cln ,+ where A, B and C are real.(7)
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
PhysicsAndMathsTutor.com June 2012
Leave blank
15
*P40111A01532* Turn over
Question 5 continued
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
PhysicsAndMathsTutor.com June 2012
Leave blank
22
*P40111A02232*
7. f ( ) cosh sinh ,x x x= −5 4 x ∈�
(a) Show that f(x) = e ex x12
9( )+ −
(2)
Hence
(b) solve f ( )x = 5(4)
(c) show that sinh
d 15 4 181
2 3
3
coshln
ln
x xx
−=⌠
⌡⎮
π
(5)
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
PhysicsAndMathsTutor.com June 2012
Leave blank
23
*P40111A02332* Turn over
Question 7 continued
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
PhysicsAndMathsTutor.com June 2012
Leave blank
16
*P42956A01632*
5. I x x x nn
n= − −∫ ( ) ,2 1 012
1
5d �
(a) Prove that, for n ��1,
( )2 1 3 5 11n I nIn nn+ = + −− ×
(5)
(b) Using the reduction formula given in part (a), find the exact value of I2 (5)
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
Leave blank
17
*P42956A01732* Turn over
Question 5 continued
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
Leave blank
24
*P42956A02432*
7.
Figure 1
The curves shown in Figure 1 have equations
y = 6 cosh x and y = 9 – 2 sinh x
(a) Using the definitions of sinh x and cosh x in terms of ex, find exact values for the x-coordinates of the two points where the curves intersect.
(6)
The finite region between the two curves is shown shaded in Figure 1.
(b) Using calculus, find the area of the shaded region, giving your answer in the form a ln b + c, where a, b and c are integers.
(6)
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
y
O x
PhysicsAndMathsTutor.com June 2013 (R)
Leave blank
25
*P42956A02532* Turn over
Question 7 continued
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
PhysicsAndMathsTutor.com June 2013 (R)
Leave blank
28
*P42956A02832*
8.
Figure 2
The curve C, shown in Figure 2, has equation
y = 2x12 , 1 ��x � 8
(a) Show that the length s of curve C is given by the equation
s = �1
8
� 1 1+⎛
⎝⎜⎞⎠⎟x
xd
(2)
(b) Using the substitution x = sinh2 u, or otherwise, find an exact value for s.
Give your answer in the form a�2 + ln(b + c�2) where a, b and c are integers.(9)
__________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
y
xO1 8
C
Leave blank
32
*P42956A03232*
Question 8 continued
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
TOTAL FOR PAPER: 75 MARKS
END
Q8
(Total 11 marks)
Leave blank
4
*P43143A0428*
2. (a) Find
∫ 14 92√ ( )x
x+
d
(2)
(b) Use your answer to part (a) to find the exact value of
−∫
3
3 14 92√ ( )x
x+
d
giving your answer in the form k ln(a + b �5), where a and b are integers and k is a constant.
(3)
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
Leave blank
8
*P43143A0828*
3. The curve with parametric equations
x = cosh 2�, y = 4 sinh �, 0 ��� ��1
is rotated through 2� radians about the x-axis.
Show that the area of the surface generated is �(cosh3 �� �� 1), where �� = 1 and � is a constant to be found.
(7)
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
Leave blank
16
*P43143A01628*
6. Given that
In =0
4
∫ x x x nn�( ) , ,16 02− d �
(a) prove that, for n ��2,
( ) ( )n I n In n+ = − −2 16 1 2
(6)
(b) Hence, showing each step of your working, find the exact value of I5(5)
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
Leave blank
17
*P43143A01728* Turn over
Question 6 continued
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
10 Edexcel AS/A level Mathematics Formulae List: Further Pure Mathematics FP3 – Issue 1 – September 2009
Further Pure Mathematics FP3 Candidates sitting FP3 may also require those formulae listed under Further Pure Mathematics FP1, and Core Mathematics C1–C4.
Vectors
The resolved part of a in the direction of b is b
a.b
The point dividing AB in the ratio μλ : is μλλμ
++ ba
Vector product: −−−
===×
1221
3113
2332
321
321ˆ sinbabababababa
bbbaaakji
nbaba θ
)()()(
321
321
321
bac.acb.cba. ×=×==×cccbbbaaa
If A is the point with position vector kjia 321 aaa ++= and the direction vector b is given by
kjib 321 bbb ++= , then the straight line through A with direction vector b has cartesian equation
)( 3
3
2
2
1
1 λ=−
=−
=−
baz
bay
bax
The plane through A with normal vector kjin 321 nnn ++= has cartesian equation
a.n−==+++ ddznynxn where0321
The plane through non-collinear points A, B and C has vector equation
cbaacabar μλμλμλ ++−−=−+−+= )1()()( The plane through the point with position vector a and parallel to b and c has equation
cbar ts ++=
The perpendicular distance of ) , ,( γβα from 0321 =+++ dznynxn is 23
22
21
321
nnn
dnnn
++
+++ γβα.
Edexcel AS/A level Mathematics Formulae List: Further Pure Mathematics FP3 – Issue 1–September 2009 11
Hyperbolic functions 1sinhcosh 22 =− xx
xxx coshsinh22sinh = xxx 22 sinhcosh2cosh +=
)1( 1lnarcosh }{ 2 ≥−+= xxxx
}{ 1lnarsinh 2 ++= xxx
)1( 11lnartanh 2
1 <−+= x
xxx
Conics
Ellipse Parabola Hyperbola Rectangular
Hyperbola
Standard Form 12
2
2
2
=+by
ax axy 42 = 12
2
2
2
=−by
ax 2cxy =
Parametric Form
)sin ,cos( θθ ba )2 ,( 2 atat (a sec θ, b tan θ ) (±a cosh θ, b sinh θ ) t
cct,
Eccentricity 1<e
)1( 222 eab −= 1=e 1>e
)1( 222 −= eab e = √2
Foci )0 ,( ae± )0 ,(a )0 ,( ae± (±√2c, ±√2c)
Directrices eax ±= ax −=
eax ±= x + y = ±√2c
Asymptotes none none by
ax ±= 0 ,0 == yx
12 Edexcel AS/A level Mathematics Formulae List: Further Pure Mathematics FP3 – Issue 1–September 2009
Differentiation
f(x) f ′(x)
xarcsin 21
1
x−
xarccos 21
1
x−−
xarctan 211x+
xsinh xcosh xcosh xsinh xtanh x2sech
xarsinh 21
1
x+
xarcosh 1
12 −x
artanh x 211x−
Integration (+ constant; 0>a where relevant)
f(x) xx d)f(
xsinh xcosh xcosh xsinh xtanh xcoshln
22
1
xa − )( arcsin ax
ax <
221
xa +
ax
aarctan1
22
1
ax − )( ln,arcosh }{ 22 axaxx
ax >−+
22
1
xa + }{ 22ln,arsinh axx
ax ++
221
xa − )( artanh1ln
21 ax
ax
axaxa
a<=
−+
221
ax −
axax
a +−ln
21
Edexcel AS/A level Mathematics Formulae List: Further Pure Mathematics FP3 – Issue 1–September 2009 13
Arc length
xxys d
dd1
2
+= (cartesian coordinates)
tty
txs d
dd
dd 22
+= (parametric form)
Surface area of revolution
Sx = sy d2π = + xxyy d
dd12
2
π
= + tty
txy d
dd
dd2
22
π
8 Edexcel AS/A level Mathematics Formulae List: Further Pure Mathematics FP1 – Issue 1 – September 2009
Further Pure Mathematics FP1 Candidates sitting FP1 may also require those formulae listed under Core Mathematics C1 and C2.
Summations
)12)(1(61
1
2 ++==
nnnrn
r
2241
1
3 )1( +==
nnrn
r
Numerical solution of equations
The Newton-Raphson iteration for solving 0)f( =x : )(f)f(
1n
nnn x
xxx′
−=+
Conics
Parabola Rectangular
Hyperbola
Standard Form axy 42 = xy = c2
Parametric Form (at2, 2at) t
cct,
Foci )0 ,(a Not required
Directrices ax −= Not required
Matrix transformations
Anticlockwise rotation through θ about O: −
θθθθ
cos sinsincos
Reflection in the line xy )(tanθ= : − θθ
θθ2cos2sin2sin 2cos
In FP1, θ will be a multiple of 45°.
Edexcel AS/A level Mathematics Formulae List: Core Mathematics C4 – Issue 1 – September 2009 7
Core Mathematics C4 Candidates sitting C4 may also require those formulae listed under Core Mathematics C1, C2 and C3.
Integration (+ constant)
f(x) xx d)f(
sec2 kx k1 tan kx
xtan xsecln
xcot xsinln
xcosec )tan(ln,cotcosecln 21 xxx +−
xsec )tan(ln,tansecln 41
21 π++ xxx
−= xxuvuvx
xvu d
ddd
dd
6 Edexcel AS/A level Mathematics Formulae List: Core Mathematics C3 – Issue 1 – September 2009
Core Mathematics C3 Candidates sitting C3 may also require those formulae listed under Core Mathematics C1 and C2.
Logarithms and exponentials xax a=lne
Trigonometric identities
BABABA sincoscossin)(sin ±=± BABABA sinsincoscos)(cos =±
))(( tantan1tantan)(tan 2
1 π+≠±±=± kBABABABA
2cos
2sin2sinsin BABABA −+=+
2sin
2cos2sinsin BABABA −+=−
2cos
2cos2coscos BABABA −+=+
2sin
2sin2coscos BABABA −+−=−
Differentiation
f(x) f ′(x) tan kx k sec2 kx sec x sec x tan x cot x –cosec2 x cosec x –cosec x cot x
)g()f(
xx
))(g()(g)f( )g()(f
2xxxxx ′−′
Edexcel AS/A level Mathematics Formulae List: Core Mathematics C2 – Issue 1 – September 2009 5
Core Mathematics C2 Candidates sitting C2 may also require those formulae listed under Core Mathematics C1.
Cosine rule
a2 = b2 + c2 – 2bc cos A
Binomial series
2
1
)( 221 nrrnnnnn bbarn
ban
ban
aba ++++++=+ −−− (n ∈ )
where )!(!
!C rnr
nrn
rn
−==
∈<+×××
+−−++×−++=+ nxx
rrnnnxnnnxx rn ,1(
21)1()1(
21)1(1)1( 2 )
Logarithms and exponentials
axx
b
ba log
loglog =
Geometric series un = arn − 1
Sn = r ra n
−−
1)1(
S∞ = r
a−1
for ⏐r⏐ < 1
Numerical integration
The trapezium rule: b
axy d ≈ 2
1 h{(y0 + yn) + 2(y1 + y2 + ... + yn – 1)}, where n
abh −=
4 Edexcel AS/A level Mathematics Formulae List: Core Mathematics C1 – Issue 1 – September 2009
Core Mathematics C1
Mensuration
Surface area of sphere = 4π r 2
Area of curved surface of cone = π r × slant height
Arithmetic series un = a + (n – 1)d
Sn = 21 n(a + l) =
21 n[2a + (n − 1)d]
top related